Presenting the fundamentals of abstract algebra to a typical class of undergraduates is easy; making them understand the subject, however, is another matter. While there is no magic formula for teaching abstract algebra, there are many tricks of the trade, and many innovations that have only been explored in the last decade or two. *Innovations in Teaching Abstract Algebra* is a veritable treasure trove of valuable ideas and should prove a useful resource for both novice and experienced mathematics teachers.

The book consists of eighteen articles, written primarily by individuals, averaging about five to ten pages in length. The articles are grouped into three sections: "Engaging Students in Abstract Algebra", "Using Software to Approach Abstract Algebra", and "Learning Algebra Through Applications and Problem Solving". Most articles don't fit neatly into any of the categories, though, as all of the authors are naturally eager to share whatever of value that they have discovered. The authors in the "Engaging Students" section tend to embrace the "active learning" teaching philosophy that has become popular recently, but the section could have been titled "General Advice". Not that any of the papers are dull or generic: this reviewer found each of the eighteen papers at least somewhat interesting and worthwhile.

One of my favorite articles is written by Steve Benson of the University of New Hampshire (with Brad Frindell). Like many (all?) college mathematics instructors today, Benson is disturbed by the tendency of many students to think that mathematics is a collections of tricks to be memorized. He writes:

One day, at the end of a class for pre-service secondary teachers, a student looked up from her desk and said, 'Wow, you sure know a lot of stuff!' I thought I had chosen activities that would show my students that they could solve problems that were new to them The young woman thought I just remembered how to do the problems — even though the problems were new to me too. It finally occurred to me that if one learns mathematics by doing mathematics (as I claimed), then perhaps the only person learning when I am up at the board is me.

Benson decided to use a discovery approach in his course. That is, the students discovered not only the proofs of the theorems, but also the key definitions and statements of the theorems themselves (wherever possible). He reports all sorts of minor difficulties he encountered, and describes the full spectrum of student reactions he received. Not surprisingly, many of his students were hostile to his teaching innovations at first; fortunately, most came to see the advantages of discovery learning by the end of the term. Amazingly, Benson writes, "my students were able to learn more course content than in traditional courses I had taught"; he attributes this to the increased interaction between teacher and students which occurred.

Many of the papers in this volume describe the author's experience teaching with a particular software package, with six packages represented in at least one paper: Finite Group Behavior (FGB), ISETL, GAP, MATLAB, Maple, and Mathematica. There is enough description of each software for an instructor to get a sense of whether he or she might profitably include the software in his or her course. The editors have thoughtfully included information on obtaining any of the software described in the book.

A typical technology paper is Kevin Charlwood's, which describes his experiences using Maple with his algebra students. "Maple is able to help our students quickly work through enough examples to give them more time to analyze the results of the computations and write about their implications," he writes. Like most of the other authors, Charlwood includes plenty of sample computer code he used with his students; instructors wishing to begin incorporating technology into their courses should find that the sample computer activities in this book to be very useful and convenient starting points. There is a permutation exercise, one of whose goals is for students to conjecture that every permutation may be expressed as a product of disjoint cycles. Charlwood also describes in detail the activities he has utilized with matrix groups and groups of complex numbers.

Few instructors are able to incorporate technology into their teaching of ring or field theory. An exception is Allen C. Hibbard, who has written his own suite of Mathematica functions for use in teaching algebra (which are available for public download). Like most algebra software, Hibbard's "AbstractAlgebra" package allows users to do extensive computations with finite groups. Hibbard's goes further allowing students to explore such topics s quotient rings and Galois fields.

I won't try to describe the other fifteen papers in this volume; suffice it to say that all are intelligent and well reasoned, so that *Innovations in Teaching Abstract Algebra* delivers eighteen insightful perspectives in one convenient source. This book is a fantastic resource for any abstract algebra teacher interested in innovative teaching techniques. I strongly recommend it.

Andrew B. Perry is Assistant Professor of Mathematics at Springfield College in Springfield, MA.